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SCIENCE CHINAMathematics. ARTICLES .February 2017 Vol.60 No.2: 277–284doi: 10.1007/s11425-016-5155-0c ? Science China Press and Springer-Verlag Berlin Heidelberg 2016 math.scichina.comMore on stability of almost surjective ε-isometries ofBanach spacesCHENG LiXin 1 , SHEN QinRui 1 , ZHANG Wen 1,∗ & ZHOU Yu 21 School of Mathematical Sciences, Xiamen University, Xiamen 361005, China;2 School of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620, ChinaEmail: lxcheng@xmu.edu.cn, 290774736@qq.com, wenzhang@xmu.edu.cn, roczhouf l y@126.comReceived December 29, 2015; accepted February 28, 2016; published online July 22, 2016Abstract Let X and Y be two Banach spaces, and f : X → Y be a standard ε-isometry for some ε > 0. Inthis paper, by using a recent theorem established by Cheng et al. (2013–2015), we show a suf f i cient conditionguaranteeing the following sharp stability inequality of f: There is a surjective linear operator T : Y → X ofnorm one so thatkTf(x) − xk 6 2ε, for all x ∈ X.As its application, we prove the following statements are equivalent for a standard ε-isometry f : X → Y :(i) liminf t→∞ dist(ty,f(X))/|t| < 1/2, for all y ∈ S Y ;(ii) τ(f) ≡ sup y∈S Y liminf t→∞ dist(ty, f(X))/|t| = 0;(iii) there is a surjective linear isometry U : X → Y so thatkf(x) − Uxk 6 2ε, for all x ∈ X.This gives an af f i rmative answer to a question proposed by Vestfrid (2004, 2015).Keywords ε-isometry, stability, sharp estimate, Banach spaceMSC(2010) 46B04, 46B20, 47A58Citation: Cheng L X, Shen Q R, Zhang W, et al. More on stability of almost surjective ε-isometries of Banachspaces. Sci China Math, 2017, 60: 277–284, doi: 10.1007/s11425-016-5155-01 IntroductionThe celebrated Mazur-Ulam theorem [21] states that every surjective isometry f from a Banach space Xto another Banach space Y with f(0) = 0 is necessarily linear. The simple example f :→ ? 2∞def i nedby f(x) = (t,sint) shows that it does not hold in general without the assumption of surjectivity on f.But the surjectivity hypothesis can be dropped if the space Y is strictly convex, or it can be weakenedas shown in [14]. In 1968, Figiel [13] showed the remarkable result “every isometry f : X → Y withf(0) = 0 admits a linear left inverse F : Y f → X of norm one”, i.e.,F ? f = Id with kFk = 1,where Y f denotes the subspace spanf(X) of Y . A generalized notion of isometry is the so-called ε-isometry: A mapping f : X → Y is said to be an ε-isometry for some ε > 0 provided|kf(x) − f(y)k − kx − yk| 6 ε, for all x,y ∈ X. (1.1)∗ Corresponding author